The authors' earlier paper presents the derivation of this general model, which considers bed dimensions, flow rates, grind size distribution, and pressure drop. They assume isothermal conditions , because optimal brewing circumstances require a narrow temperature range of 91 to 94 degrees Celsius. They also assume that coffee bed properties remain homogeneous in any cross section ,and that water saturates all pores in the coffee bed, eliminating the need to model unsaturated flow. a set of conservation equations on the bed scale monitor the transport of coffee and liquid throughout the coffee bed.

Now ,the authors take that model one step further. "The model of coffee brewing published in Chemical Engineering Science was mathematically complete, but I would describe it as a model only a computer could love: a complicated system of coupled partial differential equations ,that can only be solved numerically," Lee said. "This new paper analyses that model to produce a reduced system of equations for which approximate analytic solutions can be found."

Because coffee brewing involves so many components, simplifying the model becomes necessary. "In modelling a complicated physical process such as coffee brewing, one attempts to write down a system of equations which captures the essence of the process," O'Brien said. "In doing so, we initially make some simplifications, which neglect some aspects of the real problem. For example, real coffee contains a large number of dissolved substances; we simplify our model by considering the case of a single such substance. The mathematical model then comprises conservation laws , which in their complete form cannot be solved exactly."

The authors then utilize non-dimensionalism, which measures variables with respect to fundamental constants intrinsic to the problem, to further simplify the extraction model. This technique reduces the number of parameters, which include brew ratio, brewing time, water quality and temperature, grind size and distribution, and extraction uniformity, therefore letting the authors recognize the equations' dominant terms before they begin actively seeking solutions. "Neglecting smaller terms thus allows us to find approximate solutions," O'Brien said.